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ON THE COHERENCE CONDITIONS FOR

PSEUDO-DISTRIBUTIVE LAWS

NICOLA GAMBINO

Abstract. We survey the development of the formal theory of pseudo-monads, the analogue for pseudo-monads of the formal theory of monads.One of the main achievements of the theory is a satisfactory axiomatisationof the notion of a pseudo-distributive law between pseudo-monads.

1. Towards a formal theory of pseudo-monads

The formal theory of monads, originally introduced by Street in [22] anddeveloped further by Lack and Street [17] provides a mathematically efficienttreatment of several aspects of the theory of monads [1]. For example, it exhibitsa universal property of the category of algebras for a monad and provides aclear explanation for Beck’s axioms for a distributive law [2]. Over the pastfew years, there has been substatial progress in the development of a formaltheory of pseudo-monads [4, 15, 19, 20, 21, 23, 24], with applications to puremathematics [7, 8] and theoretical computer science [3, 5, 25]. Our aim hereis to give a survey this development, both to facilitate further applications andto provide a reference for future work. We shall focus our attention on theformulation of coherence conditions, which has proved to be one of the mostdelicate aspects of the theory.

Just as the formal theory of monads is developed within two-dimensionalcategory theory [14], the formal theory of pseudo-monads is developed withinthree-dimensional category theory [10]. Within this setting, it is convenient towork with Gray-categories, which are semistrict tricategories [10, Section 4.8].Working with Gray-categories is easier than working with general tricategories,but does not lead to an essential loss of generality, since every tricategory istriequivalent to a Gray-category [10, Theorem 8.1].

The starting point of the formal theory of pseudo-monads is the definition,for a Gray-category K, of the Gray-category PsmK of pseudo-monads in K.As we will see, this is done following a different approach to the one takenin the formal theory of monads to define the 2-category of monads in a 2-category. The change of approach allows one to avoid building into the defini-tion of PsmK the notions of a pseudo-monad morphism, pseudo-monad trans-formation, and pseudo-monad modification, which involve complex coherenceconditions. These notions can be introduced at a later stage and then shownto lead to a tricategory that is triequivalent to PsmK.

Date: July 7th, 2009.2000 Mathematics Subject Classification. 18D05,18C15,18C20.Key words and phrases. Pseudo-monad, pseudo-distributive law, coherence axioms.

1

2 N. GAMBINO

A byproduct of the formal theory of pseudo-monads that is of particularinterest for applications is the definition of a satisfactory notion of pseudo-distributive law between pseudo-monads. The notion of a pseudo-distributivelaw between pseudo-monads should be understood as an analogue of the clas-sical notion of a distributive law between monads introduced by Beck [2]. Forthis notion, the four diagrams that are required to commute in the definition ofa distributive law are replaced by diagrams commuting up to invertible 3-cells,which are then required to satisfy appropriate coherence conditions. Extendingthe set of coherence conditions for semistrict pseudo-distributive laws between2-monads introduced by Kelly [12], Marmolejo identified a set of nine coherenceconditions for pseudo-distributive laws [20]. Later, Marmolejo and Wood [21]showed not only that an additional tenth coherence condition, introduced byTanaka [23], can be derived from Marmolejo’s conditions, but also that one ofthe nine conditions originally introduced by Marmolejo is derivable from theothers, thereby reducing the axiomatization of the coherence conditions for apseudo-distributive law to eight axioms.

The first main contribution of the survey is to state precisely all the coher-ence conditions for pseudo-monads, pseudo-monad morphisms, pseudo-monadtransformations, and pseudo-monad modification, and to prove that they giverise to a tricategory that is equivalent to the Gray-category PsmK. The secondmain contribution is to give clearly all the coherence conditions for pseudo-distributive laws: the eight core conditions that are part of the definition, theninth, derivable, condition originally introduced in [20] and the tenth, deriv-able, condition stated in [23]. We also provide an interpretation of these con-ditions in terms of the notions of pseudo-monad morphisms, pseudo-monadtransformation, and pseudo-monad modification. As direct consequence of thisinterpretation, we obtain an analogue of Beck’s fundamental theorem relatingdistributive laws and liftings to categories of Eilenberg-Moore algebras [2].

We conclude these introductory remarks by recalling from [4] that the devel-opment of the formal theory of pseudo-monads in the Gray-categorical settingdoes not immediately account for the Kleisli construction. Indeed, the Kleisliconstruction for a pseudo-monad in the Gray-category 2-Cat⊗ of 2-categories,2-functors, pseudo-natural transformations, and modifications, produces a bi-category, not a 2-category, thus leading outside 2-Cat⊗. It seems therefore thata tricategorical version of the formal theory of monads, based on the existingGray-categorical theory, will eventually be needed.

2. Gray-categories

We begin by reviewing the notion of a Gray-category. Let us start with somepreliminaries. We write 2-Cat for the category of 2-categories and 2-functors.For 2-categories X and Y , let [X,Y ] be the 2-category of 2-functors from X

to Y , pseudo-natural transformations, and modifications [14]. This definitionequips the category 2-Cat with the structure of a closed category [6]. Theclosed structure of 2-Cat is part of symmetric monoidal structure, the tensorproduct of which is known as the Gray tensor product [10, Section 4.8]. Wewill write X ⊗ Y for the Gray tensor product of 2-categories X and Y . We

PSEUDO-DISTRIBUTIVE LAWS 3

write 2-Cat⊗ to emphasise that we consider 2-Cat as being equipped with theclosed symmetric monoidal structure given by the Gray tensor product.

By definition, aGray-category is a 2-Cat⊗-enriched category [10, Section 5.1].For a Gray-category K, we write K also for its set of objects. Given X,Y ∈ K,we write K(X,Y ) for the hom-2-category of maps from X to Y . We refer to theobjects of K also as 0-cells, and to the n-cells of the hom-2-categories of K asthe n+ 1-cells of K. Following this idea, every Gray-category K can be viewedas a tricategory [10, Proposition 3.1]. Gray-categories are rather special tri-categories, in that their only non-strict operation is horizontal composition of2-cells [10, Section 5.2]. For an example of a Gray-category, recall that 2-Cat⊗is a monoidal closed category and so it is enriched over itself. Therefore, itcan be viewed as a Gray-category, as we will do from now on. More explicitly,2-Cat⊗ is the Gray-category having 2-categories as 0-cells, 2-functors as 1-cells,pseudo-natural transformations as 2-cells, and modifications as 3-cells.

Let us briefly describe what the non-strictness of the horizontal compositionof 2-cells in a Gray-category amounts to. Let K be a Gray-category. Let usconsider 0-cells I,X, Y , 1-cells A,B : I → X and H,K : X → Y . The non-strictness of K means that for every pair of 2-cells f : A → B and p : H → K,we have invertible 3-cells

HA HB

KA KB

Hf//

Kf//

pA

��

pB

��

pf��

If, as usual in category theory [18, Section V.5], we think of 1-cells A,B : I → X

as generalised elements of X, then these 3-cells are analogous to the 2-cells thatare part of the structure of a pseudo-natural transformation, and indeed theysatisfy very similar coherence conditions [20, Section 2]. In the following, thesecoherence conditions will often be used implicitly.

The notions of a Gray-functor and a Gray-natural transformation are in-stances of the general notions of enriched functor and enriched natural trans-formation [13, Section 1.2]. We will use the terminology of Gray-modificationand Gray-perturbation to denote the strict counterparts of the correspondingtricategorical notions [10, Section 3.3]. When working with the Yoneda embed-ding for Gray-categories, which is just an instance of the Yoneda embeddingfor enriched categories [13, Section 2.4], we often identify an object of K withthe representable Gray-functor associated to it. Analogous conventions will beused also for the n-cells of K, where n = 1, 2, 3.

For further information on Gray-categories and tricategories, we invite thereader to refer to [9, 10, 11, 16].

3. The Gray-category of pseudo-monads

From now on, unless otherwise specified, we work with a fixed Gray-category K.

4 N. GAMBINO

Definition 3.1. A pseudo-monad (X,S) in K consists of a 0-cell X, a 1-cellS : X → X, 2-cells u : 1X → S, m : S2 → S, and invertible 3-cells

S3 S2

S2 S

mS

��

m

��

Sm //

m//

�

S S2

S

S

1S&&

Su //

m

�� 1Sxx

uSoo

λksρ

ks

satisfying the coherence axioms in (1) and (2) below:

(1)

S4 S3

S3 S2

S3

S2 S

S2m //

mS2

��

Sm //

m

��

SmS

��??

?

?

?

?

?

?

?

?

Sm

��??

?

?

?

?

?

?

?

?

mS

��mS

��??

?

?

?

?

?

?

?

?

m//

S�

4

4

4

4

4

4

4

4

α

��

αS��

=

S4 S3

S2

mm��

S3

S2 S

S2

α�

:

:

:

:

:

:

:

:

S2m //

mS

��

m

��??

?

?

?

?

?

?

?

?

?

mS2

��Sm //

m

��

Sm

��??

?

?

?

?

?

?

?

?

mS��?

?

?

?

?

?

?

?

?

?

m //

�

(2)

S2

S3 S2

S2 S

1S2

""

1S2

��

m

��

Sm

��

SuS

��??

?

?

?

?

?

?

Sm //

m//

�

Sρ��

λS��

= S2 Sm //

Note that the notion of a pseudo-monad is self-dual, in the sense that apseudo-monad in K is the same thing as a pseudo-monad in Kop , where Kop

is the Gray-category obtained from K by reversing the direction of the 1-cells,but not that of the 2-cells and 3-cells.

Proposition 3.2 (Marmolejo). Let (X,S) be a pseudo-monad in K. The fol-

lowing coherence conditions

(3)

S2 S

S3

S2 S

S2

α

� :

:

:

:

:

:

m //

uS

��

m��?

?

?

?

?

?

?

?

uS2

��Sm //

1S2

��mS

��??

?

?

?

?

?

?

m//

ρ��

um��

=

S2

S3

S2 S

uS2

��

mS��?

?

?

?

?

?

?

?

m//

1S2

��

ρS��

PSEUDO-DISTRIBUTIVE LAWS 5

(4)

1 S

S

S

S2

uu +3

ρ+3

u //

Su

��

m

��??

?

?

?

?

?

?

u

��uS //

1S

��1S //

λ +3= 1 S

u //

(5)

S2 S3

mu��

S

S2

S

S2

S2u //

Sm

��??

?

?

?

?

?

?

m

��m

��??

?

?

?

?

?

?

m

��Su //

mS

��

λks

�

1S //

=

S2 S3

S2

S

S2u //

Sm

��??

?

?

?

?

?

?

1S2 ..

m

��

Sλks

are derivable.

Proof. See [19, Proposition 8.1]. �

Let (X,S) be a pseudo-monad in K. Let us recall the definition of the2-category Ps-S-Alg(I) of I-indexed pseudo-S-algebras, pseudo-algebra mor-phisms, and pseudo-algebra 2-cells, where I ∈ K. An I-indexed pseudo-S-algebra consists of a 1-cell A : I → X, called the underlying 1-cell of thepseudo-algebra, a 2-cell a : SA → A, called the structure map of the pseudo-algebra, and invertible 3-cells

S2A

SA

SA

A

Sa //

a//

mA

��

a

��

a��

A SA

A

1A&&

uA //

a

��

a +3

called the associativity and unit of the pseudo-algebra, satisfying the coherenceaxioms (6) and (7) stated below.

6 N. GAMBINO

(6)

S3A S2A

S2A SA

S2A

SA A

S2a //

mSA

��

Sa //

a

��

SmA

��??

?

?

?

?

?

?

Sa

��??

?

?

?

?

?

?

mA

��mA

��??

?

?

?

?

?

?

a//

Sa�#

?

?

?

?

?

?

a��

αA��

=

S3A S2A

SA

ma��

S2A

SA A

SAa

� :

:

:

:

:

:

S2a //

mA

��

a��?

?

?

?

?

?

?

?

?

mSA

��Sa //

a

��

Sa

��??

?

?

?

?

?

?

mA��?

?

?

?

?

?

?

?

a//

a��

(7)

SA

S2A SA

SA A

1SA

""

1SA

��

a

��

mA

��

SuA

��??

?

?

?

?

?

?

Sa //

a//

a��

a��

λA��

= SA Aa //

Proposition 3.3 (Marmolejo). Let A be a pseudo-algebra for a pseudo-monad

(X,S). The coherence condition

(8)

SA A

S2A

SA

ua��

A

SAa

� :

:

:

:

:

:

a //

uA

��

a��?

?

?

?

?

?

?

?

?

uSA

��Sa //

1SA

mA

��??

?

?

?

?

?

?

a//

a��

=

SA

S2A

SA A

uSA

��

mA��?

?

?

?

?

?

?

?

a//

1SA

ρA��

is derivable.

Proof. See [19, Lemma 9.1]. �

As usual, we refer to a pseudo-algebra by the name of its underlying 1-cell,leaving the rest of its data implicit. Given pseudo-algebras A and B, a pseudo-

algebra morphism f : A → B consists of a 2-cell f : A → B and an invertible3-cell

SA

A

SB

B

Sf//

f//

a

��

b

��

f��

PSEUDO-DISTRIBUTIVE LAWS 7

satisfying the coherence conditions (9) and (10) stated below.

(9)

S2A S2B

SA SB

SA

A B

S2f//

mA

��

Sf//

b

��

Sa

��??

?

?

?

?

?

?

Sb

��??

?

?

?

?

?

?

a

��a

��??

?

?

?

?

?

?

?

f//

Sf

�#?

?

?

?

?

?

f��

a��

=

S2A SB

SB

SA

A B

mf��

SBf

� :

:

:

:

:

:

S2f//

mB

��

b

��??

?

?

?

?

?

?

?

mA

�� Sf//

b

��

Sb

��??

?

?

?

?

?

?

?

a��?

?

?

?

?

?

?

?

?

f//

b��

(10)

A

SA

A B

uA

��

a��?

?

?

?

?

?

?

?

?

f//

1A

��

a��

=

A B

SA

A

uf��

B

SBf

� :

:

:

:

:

:

f//

uB

��

b��?

?

?

?

?

?

?

?

?

uA

�� Sf//

1B

��a

��??

?

?

?

?

?

?

?

f//

b ��

Given pseudo-algebra morphisms f : A → B and g : A → B, a pseudo-algebra

2-cell consists of a 3-cell α : f → g satisfying the coherence condition (11).

(11)

SA

A

SB

B

Sf

''

Sg

77

g

88

a

��

b

��g

��

S�

=

SA

A

SB

B

Sf

''

f

&&

g

88

a

��

b

��

f��

�

There is a forgetful 2-functor UI : Ps-S-Alg(I) → K(I,X), defined by mappinga pseudo-S-algebra to its underlying 1-cell, which has a left pseudo-adjoint,defined by mapping a 1-cell A : I → X to the free pseudo-algebra on it, givenby the composite 1-cell SA : I → X.

The function mapping an object I ∈ K to the 2-category Ps-S-Alg(I) ex-tends to a Gray-functor Ps-S-Alg : Kop → 2-Cat⊗. We also have a Gray-transformation U : Ps-S-Alg → X, with components given by the forgetful2-functors UI : Ps-S-Alg(I) → K(I,X), for I ∈ K. Note that here we are usingthe notational conventions regarding representable Gray-functors introduced inSection 2. These conventions will be exploited repeatedly below.

We now recall from [20, Section 7] and [15, Section 6] the definition of theGray-category PsmK of pseudo-monads in K. The 0-cells are pseudo-monads(X,S) in K. For 0-cells (X,S) and (Y, T ), a 1-cell (H, H) : (X,S) → (Y, T )

8 N. GAMBINO

consists of a 1-cell H : X → Y in K and a Gray-transformation H : Ps-S-Alg →

Ps-T -Alg making the following diagram commute

Ps-S-AlgH //

U

��

Ps-T -Alg

U

��

XH

// Y

We refer to H as a lifting of H to pseudo-algebras. Analogous terminology willbe used for the notions introduced below. Given 1-cells (H, H) : (X,S) → (Y, T )

and (K, K) : (X,S) → (Y, T ), a 2-cell (p, p) : (H, H) → (K, K) consists of a

2-cell p : H → K in K and a Gray-modification p : H → K such that thefollowing diagram commutes

UHUp

//UK

HUpU

// KU

The vertical arrows are the identities given by the assumption that H and K

are liftings of H and K, respectively. Finally, for 2-cells (p, p) and (q, q), a 3-cellα : (p, p) → (q, q) consists of a 3-cell and α : p → q and a Gray-perturbationα : p → q making the following diagram commute

UpUα // Uq

pUαU

// qU

As before, the vertical arrows are the identities that are part of the assumptionthat p and q are liftings of p and q, respectively. Composition and identities ofPsmK are defined in the evident way, using those of K and 2-Cat⊗.

A Gray-category K is said to admit the construction of pseudo-algebras if forevery pseudo-monad (X,S) in K, the Gray-functor Ps-S-Alg : Kop → 2-Cat⊗is representable. When this is the case, the Yoneda lemma for Gray-categoriesimplies that the notions of liftings given above can be given an evident alter-native equivalent description, expressed purely in terms of the structure of K.Let us also recall that 2-Cat⊗ admits the construction of pseudo-algebras:the representing object for the Gray-functor Ps-S-Alg associated to a pseudo-monad (X,S) in 2-Cat⊗ is the 2-category of pseudo-algebras, pseudo-algebramorphism, and algebra 2-cells [4, 19].

4. Coherence axioms for the Gray-category of pseudo-monads

We provide an alternative description of the Gray-category PsmK, closer inspirit to the definition of the 2-categories of monads in a 2-category given in [22]and formulated without reference to the notion of pseudo-algebra. The materialin this section is essentially an account of [23, Chapter 5] and [21, Section 3],

PSEUDO-DISTRIBUTIVE LAWS 9

except for some changes in terminology, that we explain later. Theorem 4.5,however, does not seem to appear in the form given here in the existing litera-ture, even if it is closely related to [23, Theorem 5.23] and [21, Theorem 3.5].Corollary 4.6, which is inspired by [15, Section 6], seems also to be new.

Definition 4.1. Let (X,S) and (Y, T ) be pseudo-monads in K. A pseudo-

monad morphism (H,h) : (X,S) → (Y, T ) consists of a 1-cell H : X → Y , a2-cell h : TH → HS, and invertible 3-cells

T 2H

TH

THS

HS

HS2

Th //

h//

nH

��

hS��

Hm

��

h��

H TH

HS

Hu

��

vH //

h

��

h +3

These data are required to satisfy the coherence axioms in (12) and (13).

(12)

T 3H T 2HS

THS2

T 2H T 2H THS

TH HS

HS2

T 2h //

nTH

��

TnH

""DD

D

D

D

D

D

D

D

D

D

D

D

D

ThS

!!CC

C

C

C

nH

��??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

THm

""EE

E

E

E

Th//

nH

��

hS��

Hm

��

h//

⇓ T h

⇓ h

⇓ nH=

T 3H T 2HS

T 2H THS

THS2

HS2

HS3 THS

TH HS

HS2

T 2h //

nTH

��

nHS

��

ThS

""EE

E

E

E

E

E

E

Th //

nH

""EE

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

hS ""EE

E

E

E

E

E

E

hS2

��

HmS

��

HSm

""EE

E

E

E

E

E

E

THm

""EE

E

E

E

E

E

E

Hm!!D

D

D

D

D

D

D

D

D

hS

��

Hm

��

h//

⇓ nh

⇓ hS

⇓ hm

⇓ h ⇓ Hα

10 N. GAMBINO

(13)

TH

T 2H THS

HS2

TH HS

⇓λH

THu

""

1TH

��

hS

��

Hm

��

nH

��

TvH

��??

?

?

?

?

?

?

Th //

h//

⇓T h

⇓ h

=

TH

HSTHS

HS2

HS

THu

""h

��

HSu

))S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

hS

��

Hm

��

1HS

((

⇓hu

⇓Hλ

By a pseudo-monad op-morphism we mean a pseudo-monad morphism inKop .Let (X,S) and (Y, T ) be pseudo-monads in Kop . For a 1-cell H : X → Y , thedata of a 2-cell h : TH → HS and invertible 3-cells as in Definition 4.1 isreferred to as a transition from (X,S) to (Y, T ) along H in [21]. In [23], apseudo-monad morphism of the form (H,h) : (X,S) → (X,S), for a pseudo-monad (X,S), is referred to as a pseudo-distributive law of S over H.

Proposition 4.2 (Marmolejo and Wood). Let (H,h) : (X,S) → (Y, T ) be a

pseudo-monad morphism. The coherence condition

TH HS

TH HS

T 2H THS

HS2

h//

h //

vTH

��

nH

''O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

Hm

%%L

L

L

L

L

L

L

L

hS&&N

N

N

N

vHS

��Th //

HuS

��

1HS

��

⇓ vh

⇓ h

⇓Hρ

⇓ hS

=

TH HS

TH

T 2H

h//

vTH

��

nH

��??

?

?

?

?

?

?

?

?

?

1TH

��

ρH��

is derivable.

Proof. See [21, Theorem 2.3]. �

Let (H,h) : (X,S) → (Y, T ) be a pseudo-monad morphism. We show that we

can define a lifting H : Ps-S-Alg → Ps-T -Alg of H : X → Y . Let us considera fixed I ∈ K. First, let us observe that if A is an I-indexed pseudo-S-algebra,then HA is naturally an I-indexed pseudo-T -algebra, with structure map givenby the composite

THAhA

// HSAHa // HA

PSEUDO-DISTRIBUTIVE LAWS 11

and associativity and unit 3-cells provided by the pasting diagrams

T 2HA

THA

THSA

HSA

HS2A

THA

HSA

HA

ThA //

hA

//

nHA

��

hSA

��

HmA

��

THa //

HSa//

Ha//

hA

��

Ha

��

ha��

Ha��

hA��

HA THA

HA

HSAHuA ,,

1HA

**

uHA //

hA

��

Ha

��

hA +3

Ha +3

The coherence condition (6) for HA follows by an application of the coherencecondition (12) for H and the coherence condition (6) for A. The coherencecondition (7) for HA follows by applying the coherence condition (13) for H

and the coherence condition (7) for A. Secondly, we observe that if f : A → B

is a pseudo-S-algebra morphism, then Hf : HA → HB is naturally a pseudo-T -algebra morphism, as we have the following pasting diagram:

THA

HA

HSA

THB

HSB

HB

hA

��

Ha

��

THf//

HSf//

Hf//

hB

��

Hb

��

hf��

Hf��

The coherence conditions (9) and (10) follow immediately by the axioms for aGray-category. Finally, if α : f → g is a pseudo-S-algebra 2-cell, the requiredpseudo-T -algebra 2-cell is given by Hα : Hf → Hg. We have thus defined thecomponents of a Gray-natural transformation H : Ps-S-Alg → Ps-T -Alg, whichis clearly a lifting of H : X → Y .

We continue our analysis of liftings by describing what structure on a 2-cellallows us to define a lifting for it.

Definition 4.3. Let (H,h) : (X,S) → (Y, T ) and (K, k) : (X,S) → (Y, T ) bepseudo-monad morphisms. A pseudo-monad transformation (p, p) : (H,h) →

(K, k) consists of a 2-cell p : H → K and an invertible 3-cell

TH

HS

TK

KS

Tp//

pS//

h

��

k

��

p��

12 N. GAMBINO

satisfying the coherence conditions in (14) and (15) below:

(14)

T 2H T 2K

TH

THS TKS

HS2 KS2

HS KS

T 2p//

nH

��

Th

��??

?

?

?

?

?

?

?

?

?

?

Tk

��??

?

?

?

?

?

?

?

?

?

?

h

��??

?

?

?

?

?

?

?

?

?

?

?

TpS//

hS

��

kS

��pS2

//

Hm

��

Km

��

pS//

⇓T p

⇓ h⇓ pS

⇓ p−1m

=

T 2H T 2K

TH TK

TKS

KS2

HS KS

T 2p//

nH

��

nK

��

Tk

��??

?

?

?

?

?

?

?

?

?

?

Tp//

h

��??

?

?

?

?

?

?

?

?

?

?

?

k

��??

?

?

?

?

?

?

?

?

?

?

?

kS

��

Km

��

pS//

⇓np

⇓ k

⇓ p

(15)

H K

TH

HS KS

vH

��

p//

Ku

��h

##G

G

G

G

G

G

G

G

G

G

pS//

Hu

��

h�� p−1u��

=

H K

TH

HS KS

TKp

�%D

D

D

D

D

D

D

D

p//

vK

��

k##G

G

G

G

G

G

G

G

G

G

vH

�� Tp//

Ku

��h

##G

G

G

G

G

G

G

G

G

G

pS//

k��

vp��

Let (p, p) : (H,h) → (K, k) be a pseudo-monad transformation. We show

that we can define a lifting p : H → K of p : H → K, where H and K

are the liftings of H and K associated to the pseudo-monad morphisms (H,h)and (K, k), respectively. Let I ∈ K. We need to define a pseudo-natural

transformation p : HI → KI . We define the component of p associated to an I-indexed pseudo-S-algebra A to be the I-indexed pseudo-T -algebra morphismgiven by pA : HA → KA and the 2-cell

THA

HA

HSA

TKA

KSA

KA

hA

��

Ha

��

TpA//

pSA

//

pA//

kA

��

Ka

��

pA��

p−1a��

To prove the condition (9) for the pseudo-algebra morphism pA, we apply theaxioms for a Gray-category and then condition (14) for the pseudo-monad trans-formation p. To establish condition (10), it is sufficient to apply the coherencecondition (15) for the pseudo-monad transformation p, and then the axioms fora Gray-category. Clearly, p is a lifting of p as required.

PSEUDO-DISTRIBUTIVE LAWS 13

Finally, we describe what property a 3-cell has to satisfy in order to admit alifting.

Definition 4.4. Let (p, p) : (H, H) → (K, K), (q, q) : (H, H) → (K, K) bepseudo-monad transformations. A pseudo-monad modification α : (p, p) → (q, q)is a 3-cell α : p → q satisfying the coherence condition (16).

(16)

TH

HS

TK

KS

Tp

((

Tq

66

qS

66

h

��

k

��q

��

T�

=

TH

HS

TK

KS

Tp

((

pS

((

qS

66

h

��

k

��

p��

�

Given a pseudo-monad modification α : (p, p) → (q, q) we can define a liftingα : p → q of α as the Gray-perturbation with components given by the 3-cells αA : pA → qA, for a pseudo-S-algebra A. It suffices to check that, these3-cells are a pseudo-T -algebra 2-cells. To prove this, apply the axioms for aGray-category and the coherence axiom (16).

Given two pseudo-monads (X,S) and (Y, T ) in K, we define K(

(X,S), (Y, T ))

to be the 2-category having pseudo-monad morphisms from (X,S) to (Y, T ) as0-cells, pseudo-monad transformations as 1-cells, and pseudo-monad modifica-tions as 2-cells. The development in this section shows that we have a 2-functor

F(X,S),(Y,T ) : K(

(X,S), (Y, T ))

−→ PsmK

(

(X,S), (Y, T ))

.

Theorem 4.5 below can be read as saying that the coherence axioms for pseudo-monad morphisms, pseudo-monad transformations, and pseudo-monad modifi-cations are not only sufficient, but also necessary in order to obtain liftings.

Theorem 4.5. For every pair of pseudo-monads (X,S) and (Y, T ) in K, the

2-functor F(X,S),(Y,T ) : K(

(X,S), (Y, T ))

→ PsmK

(

(X,S), (Y, T ) is a pseudo-

equivalence.

Proof. Let us begin by considering a lifting H : Ps-S-Alg → Ps-T -Alg of a 1-cellH : X → Y . By the definition of a lifting, the following diagram of 2-categoriesand 2-functors commutes:

Ps-S-Alg(X)HX //

UX

��

Ps-T -Alg(X)

UX

��

K(X,X)K(X,H)

// K(X,Y )

Let us now observe that S : X → X can be regarded as an X-indexed pseudo-S-algebra, with structure map given by the 2-cell m : S2 → S. By the com-mutativity of the diagram above, this pseudo-S-algebra is mapped by the 2-functor HX into a pseudo-T -algebra with underlying 1-cell HS : X → Y , with

14 N. GAMBINO

structure map a 2-cell of the form h′ : THS → HS, and invertible 3-cells fittingin the diagrams

T 2HS

THS

THS

HS

Th′

//

h′

//

nHS

��

h′

��

h′

��

HS THS

HS

1HS))

vHS//

h′

��

h′

+3

The desired pseudo-monad morphism (H,h) : (X,S) → (Y, T ) is then obtainedby letting h : TH → HS be the composite

THTHu // THS

h′

//// HS

The appropriate 3-cells are provided by the following pasting diagrams

THS

T 2HS

TH

T 2H THS

HS

THS2

HS2

T 2Hu //

THu//

nH

��

nHS

��

Th′

//

h′

//

h′

%%

THuS //

h′S

��

Hm

��

h′

��nHu��

�

H

HS

TH

HS

THS

Hu

��

vH //

1HS 00

vHS //

THu

��

h′

��

vHu +3

h′

+3

where γ is the inverse to the 2-cell obtained from the following pasting of in-vertible 2-cells:

THS

THS2 THS

HS2 HS

1THS

��

h′

��

h′S

��

THuS

��THm //

Hm//

H�

THρ��

Let us now consider a lifting (p, p) : (H, H) → (K, K) of a 2-cell p : H → K. Wecan define a pseudo-monad transformation p : (H,h) → (K, k) by consideringthe following pasting diagram:

TH

HS

THS

TK

TKS

KS

THu

��

h′

��

Tp//

TpS//

pS//

TKu

��

k′

��

Tp−1u��

pS��

PSEUDO-DISTRIBUTIVE LAWS 15

in which the bottom 3-cell is part of the structure making pS : HS → KS intoa pseudo-algebra morphism. Finally, if (α, α) : (p, p) → (q, q) is a lifting of a3-cell α : p → q, then α : p → q is a pseudo-monad modification. Lengthycalculations show that these definitions determine a 2-functor

G(X,S),(Y,T ) : PsmK

(

(X,S), (Y, T ))

−→ K(

(X,S), (Y, T ))

which provides the required quasi-inverse to F . We omit the construction of therequired invertible pseudo-natural transformations η : 1 → GF and ε : FG → 1,since this is not difficult. �

Corollary 4.6. For every Gray-category K, there exist a tricategory K having

pseudo-monads in K as 0-cells, pseudo-monad morphisms as 1-cells, pseudo-

monad transformations as 2-cells, and pseudo-monad modifications as 3-cells,and a triequivalence F : K → PsmK.

Proof. Theorem 4.5 allows us to apply the lemma on transport of structurein [10, Section 3.6]. �

It would be of interest to define a tricategory K as in Corollary 4.6 withoutreference to the Gray-category PsmK and to verify whether this is indeed onlya tricategory, and not a Gray-category, as anticipated in [15, Section 6].

Remark. For a pseudo-monad (X,S) in K, there is a Gray-natural family ofisomorphisms of 2-categories

Ps-S-Alg(I) ∼= K(

(I, 1I ), (X,S))

for I ∈ K, where (I, 1I) denotes the identity pseudo-monad on I. Hence, Kadmits the construction of pseudo-algebras if and only if for every pseudo-monad(X,S) in K there exists an object XS ∈ K and a pseudo-monad morphism

εS :(

XS , 1XS

)

→ (X,S)

such that, for every I ∈ K, the pseudo-functor

K(

I,XS)

→ K(

(I, 1I), (X,S))

defined by composition with εS is an isomorphism. More explicitly, to give apseudo-monad morphism εS as above is to give a morphism U : XS → X, atransformation u : SU → U , and invertible modifications

S2U

SU

SU

U

Su //

u//

mU

��

u

��

u��

U SU

U

1U&&

uU //

a

��

u +3

satisfying the coherence conditions for a pseudo-monad morphism.

16 N. GAMBINO

5. Pseudo-distributive laws

Definition 5.1. Let (X,S) and (X,T ) be pseudo-monads in K. A pseudo-

distributive law of T over S consists of a 2-cell d : ST → TS and invertible3-cells

S2T

ST

STS

TS

TS2

Sd //

d//

mT

��

dS

��

Tm

��

m��

T ST

TS

Tu

��

uT //

d

��

u +3

ST 2

T 2S

ST

TS

TST

Sn //

nS//

dT

��

d

��

Td

��

n��

S

ST

TSvS

//

Sv

BB

d

��

v��

satisfying the coherence conditions (D1)-(D8) stated in Appendix A.

Proposition 5.2 (Marmolejo and Wood). Let d : ST → TS be a pseudo-

distributive law. The coherence conditions (D9) and (D10), as stated in Appen-

dix A, are derivable.

Proof. See [21, Proposition 5.1] �

The development in Section 4 allows us to give a clear explanation for thecoherence conditions for pseudo-distributive laws and for Proposition 5.2. Theaxioms (C1) and (C2) express that (T, d) : (X,S) → (X,S) is a pseudo-monadmorphism. Hence, it clear that they imply (C9), since this is a special case ofProposition 4.2. Dually, (C7) and (C8) express that (S, d) : (X,T ) → (X,T )is a pseudo-monad op-morphisms. Hence, by a dual of Proposition 5.2, asstated in [21, Proposition 4.2], they imply (C10). Let us also note that theaxioms (C3) and (C4) express that (n, n) : (T, d)2 → (T, d) is a pseudo-monadtransformation; the axioms (C5) and (C6) express that (v, v) : (X, 1X ) →

(T, d) is a pseudo-monad transformation; and finally the axioms (C7), (C8),and (C10), express that α, ρ, and λ, and are pseudo-monad modifications,respectively. It is then clear that giving a pseudo-distributive law of T over S

is equivalent to giving a lifting of T to Ps-S-Alg, by which we mean a lifting ofall the data that is part of the pseudo-monad T , thus obtaining an analogue ofBeck’s fundamental result on the equivalence between giving a distributive lawof a monad T over a monad S and a lifting of the monad T to the category ofEilenberg-Moore algebras for S [2].

PSEUDO-DISTRIBUTIVE LAWS 17

Acknowledgements

The support of the Laboratoire de Combinatoire et Informatique Mathema-tique of the Universite du Quebec a Montreal and of the Centre de RecercaMatematica is gratefully acknowledged. I would also like to thank Tom Fiore,Claudio Hermida, Martin Hyland, Steve Lack, John Power, and Mark Weberfor helpful discussions.

Appendix A. Coherence conditions for pseudo-distributive laws

We limit ourselves to drawing the boundaries of these diagrams and explainin text which 3-cells should be inserted in them, except from the 3-cells comingfrom the structure of a Gray-category of K.

(C1)

S3T S2TS

STS2

S2T

S2T STS

ST TS

TS2

S2d //

mST

��

SmT

��??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

SdS

��??

?

?

?

?

?

?

mT

��??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

STm

��??

?

?

?

?

?

?

Sd //

mT

��

dS

��

Tm

��

d//

=

S3T S2TS

S2T STS

S2T

TS2

TS3 STS

ST TS

TS2

S2d //

mST

��

mTS

��

SdS

��??

?

?

?

?

?

?

Sd //

mT

��??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

dS��?

?

?

?

?

?

?

?

dS2

��

TmS

��

TSm

��??

?

?

?

?

?

?

STm

��??

?

?

?

?

?

?

Tm

��??

?

?

?

?

?

?

dS

��

Tm

��

d//

18 N. GAMBINO

In (C1), the left-hand side pasting is obtained using Sm, m, and the associa-tivity of the pseudo-monad S; the right-hand side pasting is obtained using theassociativity of the pseudo-monad S and m.

(C2)

ST

S2T STS

TS2

ST TS

STu

""

1ST

��

dS

��

Tm

��

mT

��

SuT

��??

?

?

?

?

?

?

Sd //

d//

=

ST

TSSTS

TS2

TS

STu

""d

��

TSu

))S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

dS

��

Tm

��

Id

((

In (C2), the left-hand side pasting is obtained using Su, m, and the left unitof the pseudo-monad S; the right-hand side pasting is obtained using the leftunit of the pseudo-monad S.

(C3)

S2 S2T

S

STS

TS2

TS

S2v //

m

��

Sd

��??

?

?

?

?

?

?

SvS ..

vS2

++

vS ..

Tm

��

dS

��

=

S2 S2T

S

STS

TS2

TS

ST

S2v //

Sd

��??

?

?

?

?

?

?

dS

��

Tm

��

d

��??

?

?

?

?

?

?

?

m

��Sv //

mT

��

vS ..

For (C3), the left-hand side pasting is obtained using Sv, vS; the right-handside is obtained using m and m.

PSEUDO-DISTRIBUTIVE LAWS 19

(C4)

1X T

S ST

TS

v //

uT

��

d��?

?

?

?

?

?

?

?

?

u

��Sv //

Tu

��vS,,

=

1X T

S

TS

v //

Tu

��

u

��

vS,,

For (C4), the left-hand side pasting is obtained using u and v.

(C5)

S2T 2 S2T

TS2T

STST

ST 2S STS

T 2S2 TS2

TSTS

ST 2

TST

T 2S TS

S2n //

mT 2

��

SdT

��??

?

?

?

?

STd

��??

?

?

?

?

Sd

��??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

SnS //

dTS

��

TdS

��

dS

��nS2

//

dST

��

TmT

��

TSd ��??

?

?

?

?

T 2m

��

Tm

��

dT ��??

?

?

?

?

Td ��??

?

?

?

?

nS//

=

S2T 2 S2T

STSTS

TS2

ST 2

T 2S TS

TST

S2n //

Sd

��??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

dS��

Tm

��

mT 2

��

dT ��??

?

?

?

?

Sn //

Td ��??

?

?

?

?

nS//

mT

��

d

��??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

20 N. GAMBINO

For (C5), the left-hand side pasting is obtained using Sn, nS and mT ; theright-hand side pasting is obtained using m and mut.

(C6)

T 2 T

ST 2 ST

TST

T 2S TS

n //

Sn //

nS //

uT 2

��

uT

��

dT��

4

4

4

4

4

4

4

Td

��4

4

4

4

4

4

4d

��4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

4

Tu

=

T 2 T

ST 2

TST

T 2S TS

uT 2

��

Td��

4

4

4

4

4

4

4

n //

Tu

T 2u

dT��

4

4

4

4

4

4

4

nS//

TuT

��

In (C6), the left-hand side pasting is obtained using u, and n; the right-handside pasting is obtained using uT .

To state the coherence conditions (D8), (D9) and (D10), let α, λ, and ρ bethe associativity, left unit, and right unit for the pseudo-monad T .

(C7)

ST 3 ST

TST 2

T 2ST

T 3S TS

ST 2

S2T

TST

T 2S

dT 2

��

TdT

��

T 2d

��

d

��

dT

��

Td

��

STn 11 Sn

##

SnT--

nST,,

nTS--

Sn

;;

nS

;;

=

ST 3 ST

TST 2

T 2ST

T 3S TS

ST 2

TST

T 2S

T 2S

dT 2

��

TdT

��

T 2d

��

d

��

dT

��

Td

��

STn 11 Sn

##

TSn 22

TnS 11

nTS--

nS

##

nS

;;

For (C7), the left-hand side pasting is obtained using Sα, n, nT ; the right-handside pasting is obtained using n and αS.

PSEUDO-DISTRIBUTIVE LAWS 21

(C8)

ST ST

TS TS

ST 2

TST

T 2S

d

��

d

��

dT

��

Td

��

1ST

$$

SvT--

vST

��

vTS--

Sn;;

nS

;;

=

ST

TS TS

T 2S

d

��

1TS

$$

vTS--

nS

;;

For (C8), the left-hand side pasting is obtained using Sρ and n, vS; the right-hand side pasting is ρS.

(C9)

ST TS

ST TS

S2T STS

TS2

d//

d //

uST

��

mT

''O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

Tm

%%L

L

L

L

L

L

L

L

dS

&&N

N

N

N

uTS

��Sd //

TuS

��

1TS

��

=

ST TS

ST

S2T

d//

uST

��

mT

��??

?

?

?

?

?

?

?

?

?

?

1ST

��

For (C9), the left-hand side pasting is obtained using the right unit of thepseudo-monad S, u; the right-hand side pasting is obtained using the right unitof the pseudo-monad S.

22 N. GAMBINO

(C10)

ST ST

TS

ST 2

d

��

STv 11 Sn

##

1ST

::

=

ST ST

TS TS

ST 2

TST

T 2S

d

��

d

��

dT

��

Td

��

STv 11 Sn

##

TSv

@@

TvS

11

1TS

::

nS ##

For (C10), the left-hand side pasting uses Sλ. The right-hand side pasting isobtained using n and λS.

References

[1] M. Barr and C. Wells. Toposes, triples, and theories. Springer, 1985.[2] J. Beck. Distributive laws. In Seminar on Triples and Categorical Homology Theory

(ETH, Zurich, 1966/67), pages 119–140. Springer, 1969.[3] G. L. Cattani and G. Winskel. Profunctors, open maps, and bisimulation. Mathematical

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the author’s web page.[6] S. Eilenberg and G. M. Kelly. Closed categories. In Proceedings of La Jolla Conference

on Categorical Algebra, pages 421–562. Springer-Verlag, 1966.[7] M. Fiore, N. Gambino, M. Hyland, and G. Winskel. The cartesian closed bicategory of

generalised species of structures. Journal of the London Mathematical Society, 77(2):203–220, 2008.

[8] R. Garner. Polycategories via pseudo-distributive laws. Advances in Mathematics,218(3):781–827, 2008.

[9] R. Garner and N. Gurski. The low-dimensional structures formed by tricategories. Math-ematical Proceedings of the Cambridge Philosophical Society, 146(3):551–589, 2009.

[10] R. Gordon, A. J. Power, and R. Street. Coherence for tricategories. Memoirs of theAmerican Mathematical Society, 117(558), 1995.

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Universita di Palermo & University of Manchester

E-mail address: nicola.gambino@gmail.com